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Proceedings of the Hamburg International Conference of Logistics (HICL) – 28 Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and Jorin Dornemann Fleet Based Schedule Optimisation for Product Tanker Considering Shipʼs Stability Published in: Digital Transformation in Maritime and City Logistics Carlos Jahn, Wolfgang Kersten and Christian M. Ringle (Eds.) September 2019, epubli CC-BY-SA 4.0

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Page 1: Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and ... · Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and Jorin Dornemann Fleet Based Schedule Optimisation for

Proceedings of the Hamburg International Conference of Logistics (HICL) – 28

Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and Jorin Dornemann

Fleet Based Schedule Optimisation forProduct Tanker Considering ShipʼsStability

Published in: Digital Transformation in Maritime and City Logistics Carlos Jahn, Wolfgang Kersten and Christian M. Ringle (Eds.)

September 2019, epubliCC-BY-SA 4.0

Page 2: Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and ... · Anisa Rizvanolli, Alexander Haupt, Peter Marvin Müller and Jorin Dornemann Fleet Based Schedule Optimisation for

Keywords: Cargo Scheduling in Tramp Shipping, Optimization, Ship Stability,

Mixed Integer Linear Programming

First received: 20.May.2019 Revised: 29.May.2019 Accepted: 19.June.2019

Fleet Based Schedule Optimisation for Product Tanker Considering Ship’s Stability Anisa Rizvanolli1, Alexander Haupt2, Peter Marvin Müller1,

Jorin Dornemann2

1 – Fraunhofer Center for Maritime Logistics and Services

2 – Hamburg University of Technology

Purpose: Scheduling a fleet of product tankers in a cost effective and robust way to satisfy orders is a complex task. A variety of constraints and preferences complicate this attempt. Manual solutions as common in tramp shipping are not sufficient to deliver optimal and robust schedules. Methodology: For this, we present a mixed integer linear programming formulation of the scheduling problem. Additionally intact stability calculations for each ship of the fleet are implemented in a separate program that checks the feasibility of MILP solutions and creates new cuts for the integer program. Findings: Usually the checking of the stability criteria is done before an order will be accepted and the schedule of the ship is planed accordingly. This requires the selec-tion of a ship a priori. Checking the admissibility of a voyage gives access to a wider variety of possible combinations. Originality: To our knowledge fleet scheduling under consideration of intact stabil-ity requirements has received little attention in the literature. Previous works make very simple assumptions on the capacity of the ships and do not include in their lin-ear pro- grams any stability models.

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396 Anisa Rizvanolli et al.

1 Introduction

Shipping even if not obvious is one of the most important industries with a

big impact in our everyday life. Around 90% of the world trade is trans-

ported by the shipping industry (Ronen, 2019). Deep-sea shipping is the

only way for transportation of a large amount of goods over the world. For

many countries shipping belongs to one of their key industries. The finan-

cial crisis 2007 has changed this industry drastically. The reduction of profit

margins on one side and the vessel as a huge capital investment for a ship-

ping company with thousands of dollars daily total operation costs on the

other side have increased the focus on optimization for cost savings. The

efficiency of fleet schedules has a significant financial impact for the com-

pany.

Tramp shipping as described in (Lawrence, 1972) is one of the three main

modes of maritime operations which include additionally industrial and

liner shipping. Tankers usually do not follow fixed itineraries as container

ships. They travel according to customers’ orders and needs and are very

often compared with taxi services. Shipping companies have long- term

contracts as well as spot cargoes. The main aim of shipping companies

working in this mode is to maximize their profits by appropriately schedul-

ing and routing the fleet for the long-term contracts which implicitly limit

the choice of spot cargoes.

Scheduling a fleet of product tankers in a cost effective and robust way to

satisfy as many orders as possible is a complex task. This challenge can be

defined as the problem of allocating the right assets to the right cargoes,

such that all orders are delivered on time, at the right ports and by taking

into account customer preferences as well as legislative requirements.

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Fleet Based Schedule Optimisation for Product Tanker 397

These requirements vary from minimum age of the ship to its vetting rec-

ords. A robust scheduling and routing of the long term contracts has a di-

rect impact on the online scheduling problem of assigning spot cargoes to

ships. In this paper the focus will be on scheduling long-term orders in an

optimal way under consideration of cargo, port and especially fleet and

ship stability requirements. This problem is also known as the problem of

cargo routing based on the classification and descriptions made in (Al-

Khayyal and S.-J. Hwang, 2007). In contrast to this, in the inventory sched-

uling problem the cargo demand is determined by the schedules.

Nowadays in the shipping companies the very time consuming task of fleet

scheduling is done manually and based on planners’ experience. This pro-

cess as described in (Trottier and Cordeau, 2019) and also confirmed during

an interview with a tanker shipping company from Hamburg is a very itera-

tive, manual and difficult one. Basically, the planners try to assign the man-

datory orders first and then fill up the gaps with spot cargoes in a profitable

way.

Obviously it is impossible even for a very small fleet to create optimal

schedules in a manual way such that all constraints and requirements are

taken into consideration. The suboptimal manual solutions lead to dead

locks in the assignments of orders due to the lack of robustness or to a rel-

atively high number of idle days and therefore to considerable financial

loss. Safety and security are relevant for all mode of maritime transporta-

tion, but specially tanker ships have to fulfill high standards and require-

ments in security, which affect directly the company operations. Ship sta-

bility is at this point of high relevance and needs to be considered when

optimizing the schedules in order to guarantee safety and save costs and

reputation loss due to accidents. Nowadays this process is done also in a

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398 Anisa Rizvanolli et al.

manual way and involves the engineering department that checks individ-

ual solutions proposed by the chartering department and revokes them if

the stability of the ship is not guaranteed. The main aim is to schedule a

given fleet in such a way that it delivers as much cargo as possible as a

whole by maximizing the profit and on the same time complying with all

cargo and ships stability requirements and restrictions. The problem is for-

mulated as a mixed integer linear problem (MILP) with lazy constraints be-

ing generated by a separate module that calculates the stability of each

ship given the cargo load from the MILP, as shown in Figure 1. Including this

step in the optimization leads to more realistic results then existing MILP

formulations, that are limited only to draft restrictions for ports and do not

take ship design into consideration. Numerical experiments have shown

that a large number of feasible solutions of the MILP do not comply with the

stability requirements. A detailed description of the problem with all con-

straints and the mathematical formulation for the MILP and ships stability

calculation can be found in section 3.

Figure 1: Program interaction

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Fleet Based Schedule Optimisation for Product Tanker 399

This paper is organized as follows. In section 2 the relevant literature is pre-

sented. The problem is described in detail in section 3. In section 4 first nu-

merical experiments are presented and their results are analyzed. Conclud-

ing remarks and future work can be found in section 5.

2 Literature Review

Optimization models and methods for ship routing and scheduling prob-

lems are relatively new for this old and traditional transportation industry.

Thus problems are considered to be complex and not easy to classify and

solve. In comparison to air, train or public transportation shipping is char-

acterized by a high uncertainty in the voyage and cargo information, which

limits the usage of deterministic models. Nevertheless in the last years a

significant growth in the number of scientific publications in this field has

been recognized. The problem of ship routing and scheduling presented in

this paper is considered as a special case of the general vehicle routing

problem, see (Desrosiers et al. 1995). Methods and formulations used for

this class of problems can not be overtaken for the cargo scheduling prob-

lems in tramp shipping due to the special properties of maritime transpor-

tation including the fact that ships operate 24 hours a day and have special

physical and legislative constraints. General surveys on ship routing and

scheduling can be found in (Ronen, 1983 and Christiansen, Fagerholt, and

Ronen, 2004). In (Ronen, 1983) one of the first detailed classification

schemes for ship routing and scheduling problems is introduced. This pa-

per presents also a first analysis of the differences between the vehicle and

ship scheduling and routing problems. It includes almost all modes of ship-

ping like short and long-term as well as liner, tramp and industrial shipping

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400 Anisa Rizvanolli et al.

and further criteria that support the classification of the problem and the

choice of the appropriate solution methods. The proposed classification

schema focuses on the cost optimization and includes only port entry con-

straints for the ship and no further ship design or stability constraints. Same

holds for the classification schema presented in later papers as for example

(Christiansen, Fagerholt, and Ronen, 2004). One of the latest general survey

papers is (Christiansen, Fagerholt, Nygreen, et al., 2013). (Appelgren, 1969)

and (Appelgren, 1971) are two of the first papers dealing with the problem

of the optimal assignment of cargoes to a given fleet under consideration

of all tramp shipping specific requirements. In both papers the author pre-

sents a mixed integer linear program formulation of the scheduling prob-

lem and uses Dantzig-Wolfe decomposition and branch and bound to solve

small instances of the problem. In the following years the formulation of

the cargo routing and scheduling problem has evolved by including more

parameters and making the models more realistic as it is being stated in

(H.S. Hwang, Vi- soldilokpun, and Rosenberger, 2008). Varying the ship’s

speed and using it as a decision variable (Norstad, Fagerholt, and Laporte,

2011) has lead to significant changes in the results and cost saving (Fager-

holt et al., 2013). On the other side the introduction of this variable has in-

creased the complexity of the MILPs as well as the solution time. Beside this

sailing much slower than service speed may look good in theory but often

forces the ship to put the helm at big angles which causes a major increase

in resistance and thus implifying an increase in operational cost and risk of

casualty. For solving large and real world data instances different problem

specific heuristic methods like the ones proposed in (Malliappi, Bennell,

and Potts, 2011 and Jetlund and Karimi, 2004) as well as known local search

methods including tabu search, as presented in (Korsvik, Fagerholt, and

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Fleet Based Schedule Optimisation for Product Tanker 401

Laporte, 2010), have been developed. A further parameter being consid-

ered in later works as in (Brønmo et al., 2007) is the parcel size, which in

most cases is considered as fixed. In this work a genetic algorithm that

takes advantage of the problem’s specific structure and properties is imple-

mented and enables the solution of large data instances within reasonable

computation time. In (Cóccola and Méndez, 2015) a combination of a small

instance MILP and a heuristic that delivers near-optimal solutions for large-

scale cargo scheduling problems is presented. The focus of the mathemat-

ical model lies in costs minimization and the constraints do not include any

ship design requirements from a stability point of view. The benchmark pa-

per (Hemmati et al., 2016) gives a state of the art overview of the formula-

tions for cargo routing and scheduling problems and proposes a bench-

mark suite for different real world size problems. It is one of the first papers

dealing with the lack of standardized data for scheduling problems in

tramp shipping and proposing data for benchmarks. In this paper a modern

implementation of the adaptive large neighbourhood search (ALNS) heu-

ristic method is presented which has been developed firstly by (Ropke and

Pisinger, 2006). One of the latest papers (Homsi et al., 2018) presents a

branch and price algorithm as an exact method and a hybrid metaheuristic

for large instances considering 50 ships and 130 cargoes. To our knowledge,

in actual existing formulations and proposed solution methods for the

cargo scheduling problem in tramp shipping none of them are really con-

sidering the stability or the design of the ships as an integral constraint of

the MILP. In (Fagerholt et al., 2013) the stability of ships that transport very

special cargo is mentioned as a constraint to be taken into consideration. It

is checked manually by the engineering department which can revoke the

solution if the ship is not stable with the given load.

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402 Anisa Rizvanolli et al.

3 Problem Description and Formulation

The main purpose of this paper is to present a MILP formulation of the cargo

scheduling problem in tramp shipping under consideration of ships stabil-

ity constraints. The problem considers long-term orders to be charged and

discharged in different ports within a given time frame. Each order has its

own profit value. The fleet of tankers is homogeneous and each ship has a

different design and number of cargo tanks. Compatibility constraints for

different cargo types are taken into account. For safety reasons not all

chemicals can be charged to adjacent tanks as well as in succession in the

same tank. These requirements are formulated as constraints in the MILP.

Cargo can also be charged and discharged between ships. In maritime

terms, this is called ship-to-ship operation. The velocity of the ships is as-

sumed to be constant.

The assignment of cargo to a ship from the fleet is checked for intact stabil-

ity based on the ships design, its cargo hold configuration and the filling

level of each tank. The rules from (IMO, 2002) that apply to tankers are

checked by taking free surfaces of the liquid cargo into account. This means

that the solution of the MILP solver could assign only partially filled tanks

what represents common practice in tanker operation. Because the free

surface of a fluid within any tank aboard causes huge reduction of the sta-

bility of the ship the model for stability calculation has to be precise. The

gain one obtains is a much greater set of possible solutions for the fleet op-

timisation problem.

The objective function of the problem consist of finding the most profitable

assignment of cargoes to the ship tanks compliant with the ships stability

requirements and time constraints of the orders. The interplay between the

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Fleet Based Schedule Optimisation for Product Tanker 403

stability checker and the MILP leads to reduction of the feasible region. Nev-

ertheless the problem is very hard to solve and even small data instances

take a very long time to achieve an optimal solution with a zero gap. In this

section the problem formulation as a MILP and the stability calculations are

presented in detail.

3.1 Integer Program The aim of this integer program is to determine the fleet coordination while

maximizing the number of fulfilled orders and while minimizing the costs,

which arise from loading and unloading cargo in ports and from the ship

travelling from one port to another.

Parameters λspq The cost per journey of a ship s from port p to q. μsp The cost per loading process of ship s in port p.

P The set of all ports.

S The set of all ships.

T The time horizon.

Y The set of all types of goods.

C The set of all tanks.

O The set of all orders.

csize The size of tank c.

ssize The size of ship s.

τ The loading time, i.e. how long a ship has to be in a port

to finish the loading (same for all ships) Cs The set of all tanks on ship s. γzy is one, if good 𝑦 ∈ Y cannot be carried next to good z ∈ Y

on a ship, otherwise zero.

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404 Anisa Rizvanolli et al.

L

Ac The set of all tanks, which are adjacent to tank c. δzy is 1, if good 𝑦 can be carried immediately after good z in

a tank.

PBo The pickup begin of order o. Similar for PE (pickup end),

DB (delivery begin) and DE (delivery end). amounto The amount of the good, which is ordered in order o. ro The amount of money which is earned when fulfilling or-

der o. typeo The type of good which is ordered in order o. αspq The time it takes ship s to travel from p to q. sc is the ship s which carries tank c. C the set of all tanks on ship s which have some sort of

loading in a given solution.

Variables

pspt is 1 if ship s is at port p at time t.

ℓspt is 1 if ship s is at port p at time t and just finished loading. aspqt is 1 if ship s just ended a cruise from port p to port q at t. hcyt is the amount of type 𝑦 tank c contains at time t. bcyt is 1 if tank c contains at least 1 unit of type 𝑦 at time t. xcyat is 1 if tank c contains exactly the amount a of type 𝑦 at t. fsypt is the amount of type 𝑦 which flows from ship s to port p

at t. got the amount of order o to be picked up at time t. dot the amount of order o delivered at time t. eo is one if and only if order o is completed.

S

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Fleet Based Schedule Optimisation for Product Tanker 405

The following function represents the objective function to be used in the

problem setting. It contains two terms that represent the earning maximi-

zation and the port as well as journey cost minimization leading in total to

a profit maximization.

max 𝑒 ⋅ 𝑟∈

𝜇 ⋅ ℓ 𝜆∈

⋅ 𝑎∈

∗  

Constraints 0 𝑝 1 ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇0 ℓ 1 ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇

0 𝑎 1 ∀𝑠 ∈ 𝑆, 𝑝, 𝑞 ∈ 𝑃, 𝑡 ∈ 𝑇0 ℎ 𝑐size ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇

0 𝑏 1 ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇0 𝑥 1 ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇, 0 𝑎 𝑐size𝑠size 𝑓𝑠𝑦𝑝𝑡 𝑠size∀𝑠 ∈ 𝑆, ∀𝑦 ∈ 𝑌, ∀𝑝 ∈ 𝑃, ∀𝑡 ∈ 𝑇

0 𝑔  ∀𝑜 ∈ 𝑂, 𝑡 ∈ 𝑇0 𝑑  ∀𝑜 ∈ 𝑂, 𝑡 ∈ 𝑇

0 𝑒 1 ∀𝑜 ∈ 𝑂

(1) 

The constraints in equation 1 set the basic domains of the variables.

𝑝∈

1∀𝑠 ∈ 𝑆, 𝑡 ∈ 𝑇 (2)

Constraint 2 ensures that a ship s is at no more than one port at a time.

ℓ 0 ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 𝜏, 𝑡 ∈ 𝑇 (3)

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406 Anisa Rizvanolli et al.

Constraint 3 ensures that in the beginning no ship finishes loading before it

was in the port for at least 𝜏 time slots.

𝑝 𝜏 1 ℓ ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 𝜏 (4)

Constraint 4 ensures that the ship was in port 𝑝 for at least 𝜏 time slots be-

fore it finishes loading. Note that ℓ doesn’t have to be equal to one if the

ship was at the port for 𝜏 time slots.

𝑎𝑠𝑝𝑞𝑡 𝑝𝑠𝑞𝑡 𝑝𝑠𝑞 𝑡 1𝑝 𝑞 ∈ 𝑃

∀𝑠 ∈ 𝑆, 𝑞 ∈ 𝑃, 𝑡 1 (5)

Constraint 5 ensures that if a ship 𝑠 is in port 𝑞 in time 𝑡, it either was in port

q at time 𝑡 1 or it just arrived, i.e. ended a cruise from port p to q for some

p. 𝑎 0 ∀𝑠 ∈ 𝑆, 𝑝 𝑞 ∈ 𝑃, 𝑡 𝛼 (6)

where 𝛼 is the distance between p and 𝑞 divided by the speed of ship s.

Constraint 6 sets all variables 𝛼 equal to zero, if there is no possibility,

that a ship arrived from p at q at time t.

𝑝 𝛼 𝑎 𝛼 1 ∈

𝑝

∀𝑠 ∈ 𝑆, 𝑝 𝑞 ∈ P, 𝑡 𝛼      (7) 

If 𝛼 equals zero, constraint 7 has no impact since the left hand side is

greater or equal to 𝛼 1 and the right hand side is at most 𝛼 1

because of constraint 2. If 𝛼 = 1 (which means ship s just arrived from

a cruise from p to q), ship s must have been in port p at time

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Fleet Based Schedule Optimisation for Product Tanker 407

t 𝛼 (𝑝 = 1), because otherwise the left hand side would be

equal to minus one and the right hand side is not negative. So in this case

the left hand side is equal to zero and therefore the ship couldn’t be some-

where else other than on the cruise from p to q in the mean time, which

means each 𝑝 on the right hand side has to be equal to zero.

ℎ 𝑐size ⋅ 𝑏 ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇 (8)

Constraint 8 ensures that the variable 𝑏 is set to one if tank c is loaded

with some cargo type y at time t.

𝑏∈

1 ∀𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (9)

Constraint 9 ensures that a tank carries just one type of good at a time.

ℎ ℎ∈

𝑓∈

0

∀𝑠 ∈ 𝑆, 𝑦 ∈ 𝑌, 𝑡 1 (10)

Constraint 10 ensures that the change in the amount of good of type 𝑦 on

ship s from t to t − 1 equals the flow of good 𝑦 from ship s to some port at

time t. 𝑓 𝑠size ⋅ ℓ ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇, 𝑦 ∈ 𝑌 (11)

Constraint 11 ensures that the flow 𝑓 of type y from the ship s to port p is at most the size of the ship at each time.

ℎ ℎ 𝑐size ⋅ ℓ∈

∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 1 (12)

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408 Anisa Rizvanolli et al.

where sc denotes the ship s which carries tank c. This constraint 12 ensures

that the amount of a good y in a tank just changes if the ship is at least τ

time slots in a port to finish loading and that the change in the amount of y

in c from 𝑡 1 to t is at most the size of this tank.

𝑓∈

𝑔 𝑔:

𝑑 𝑑: o

∀𝑝 ∈ 𝑃, 𝑦 ∈ 𝑌, 𝑡 1 (13)

Constraint 13 ensures that the flow of good 𝑦 from all ships into the port at

each time t plus the change in the amount of 𝑦 which is to be picked up (i.e.

the flow of good 𝑦 from the port onto the ship) equals the change in the

amount of 𝑦 which is to be delivered. Note that fsypt can be negative, i.e. a

flow from the port onto a ship.

𝑑 𝑑 ∀𝑜 ∈ 𝑂, 𝑡 ∈ 𝑇 (14) 

𝑔 𝑔 ∀𝑜 ∈ 𝑂, 𝑡 ∈ 𝑇 (15)

Constraints 14 and 15 ensure that the amount of a delivered good 𝑦 within

an order just increases and similar the amount of a good 𝑦 to be picked up

within an order just decreases. The binary variables xcyat are introduced

solely for the reason to exclude a specific loading configuration of a ship.

With the following two constraints we set xcyat equal to 1 if and only if ex-

actly the amount a > 0 of type y is in tank c at time t:

𝑥size

𝑏 ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇 (16) 

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Fleet Based Schedule Optimisation for Product Tanker 409

𝑎𝑥size

ℎ ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇 (17) 

With these variables we are able to tell the program to forbid a specific load-

ing configuration if the ship stability constraints are hurt. Assume we have

a solution which is feasible for our program, but does not satisfy the stabil-

ity constraints. Let s be the ship which is instable when leaving port p. Our

aim is to forbid exactly the given loading configuration for ship s. 𝐶 is the

set of all tanks on ship s which have some sort of loading in our solution. Let

yc be the type which is loaded in c ∈ 𝐶 and similar ac the amount of type

yc in c. From constraints 16 and 17 we know that xcyat = 1 for y = yc and a =

ac and zero otherwise. We then are able to add a constraint which ensures

that the loading configuration for ship s is changed:

1 𝑥∈

∈ ∖

ℎ∈

𝑝 𝑝

∀𝑡 ∈ 𝑇 (18) 

The right hand side of constraint 18 just defines the time slot t at which the

ship s leaves port p, since for t the right hand side equals 1 and for all other

time slots it is at most zero. The left hand side is zero for our given solution

with which ship s is instable. To change the loading configuration we have

two options. Either we change the amount of the loading in our tanks 𝑐 ∈

𝐶 so that in the first sum of the left hand side we get that at least one xcycact

is zero. Or additional tanks are loaded so that in the second sum at least

one hcyt gets at least one.

Another problem we need to address is the specific adjacencies of tanks on

the ship. There are goods which are not allowed to be carried next to each

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410 Anisa Rizvanolli et al.

other on a ship for various safety reasons. Therefore our feasible solution

has to ensure that these adjacencies are considered. This adjacency con-

straint is constructed via the parameter 𝛾 :

𝑏 1 𝛾𝑦 𝑏𝑐𝑦∈

∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 𝑡 ∈ 𝑇 ̃∈

(19)

The right hand side of constraint 19 is less or equal zero if and only if there

is a tank adjacent to c, which is loaded with a good 𝑦 that is not allowed to

be carried next to good 𝑦. Therefore bcyt has to be equal zero, which means

that tank 𝑐 is not allowed to load good 𝑦 at time t.

Similarly there are goods which are not allowed to be carried in a tank di-

rectly after another good, because for example there have to be two sepa-

rate cleanings between carrying those two goods.

𝑏 1 𝛿 1∈

𝑏 ∀𝑐 ∈ 𝐶, 𝑦 ∈ 𝑌, 1 (20)

The right hand side of constraint 20 equals one if and only if either the tank

c didn’t carry any good at time 𝑡 1 or good y is allowed to be carried im-

mediately after the good which was in the tank at 𝑡 1. Otherwise the right

hand side equals zero and therefore bcyt has to be zero, meaning good y is

not allowed to be in tank c at time t.

The following constraints set the variable eo to one if and only if order o

is completed, which is important for our objective function: 𝑔 PEo 0 ∀𝑜 ∈ 𝑂

𝑑 DBo 0 ∀𝑜 ∈ 𝑂𝑑

o𝑎𝑚𝑜𝑢𝑛𝑡o ⋅ 𝑒 ∀𝑜 ∈ 𝑂

𝑔 PBo 𝑎𝑚𝑜𝑢𝑛𝑡o ⋅ 𝑒  ∀𝑜 ∈ 𝑂𝑔 𝑎𝑚𝑜𝑢𝑛𝑡o ⋅ 𝑒  ∀𝑜 ∈ 𝑂

(21)

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For the sake of running time we constructed a few extra constraints, which

are direct consequences of the constraints presented above, but help the

program to exclude infeasible solutions faster.

𝜏 𝑝 𝑝 𝑝 ∀𝑠 ∈ 𝑆, 𝑝 ∈ 𝑃, 𝑡 1 (22)

The statement of constraint 22 is, that if a ship arrived at a port, it stays at

this port for at least the τ time slots it takes to (un)load. This ensures that a

ship doesn’t go to a port without doing anything there.

𝑓∈

0 ∀𝑠 ∈ 𝑆, 𝑦 ∈ 𝑌 (23)

Constraint 23 is a consequence from the fact that a ship is empty in the be-

ginning and in the end, so the sum of the flow from each ship to all ports

equals zero for each good over the time span.

𝑚𝑎𝑥 0,∈

𝑎𝑚𝑜𝑢𝑛𝑡o:PBo

∧PEo ∧typeo

𝑎𝑚𝑜𝑢𝑛𝑡o:DBo

∧DEo ∧typeo

∈ ,

𝑠sizeℓ∈

 

∀𝑝 ∈ 𝑃, 𝑎 , 𝑢 ∈ 𝑇, 𝑎 𝑢 (24)

Constraint 24 ensures that for each time interval at a port the sum of the

sizes of ships that load in this port are large enough, meaning the sizes of

the ships are able to fit all orders.

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3.2 Ship stability calculation In general the stability of a ship is the ability to come back to a stable equi-

librium after any perturbation caused the ship to heel or trim. Because usu-

ally the trim does not cause safety issues this is not a matter of safety. To

make sure that all ships on duty do withstand such influences there are cer-

tain rules that apply to the ship. These rules are issued by the International

Maritime Organisation (IMO) and apply to any ship of more than 24 m

length. Before a ship will leave the harbour, the master shall ensure that the

ship is compliant to these rules for the current loading condition as well as

for the condition when arriving at the destination. Accidents due to lack of

stability lead to cost and reputation loss which has a great impact in com-

pany’s future contracts and therefore a long-term effect.

Even though the stability of a tanker ship might not be a major issue in the

first place, in particular because bulk carriers for any kind are well known

for their high stability, it is necessary to check whether a ship does fulfill the

stability criteria from the IMO, 2002 (2008 IS-Code) or it will not be allowed

to leave the harbour. Because the MILP solver does not know if a ship would

be capable of sailing at a given loading condition the stability has to be

taken into account when choosing an admissible solution for the fleet

scheduling problem.

Therefore the results of the MILP solver, where its output is the loading con-

dition of each ship of a fleet at departure, have to be evaluated. A separate

program has been written to perform stability calculations when it has

been given a ship and its loading condition as input. The output of the pro-

gram is a boolean true/false that indicates the stability of the loaded ship.

As shown below this is an admissible way to involve the requirements eval-

uation because of the interlaced quantities and the non linearity of the

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Fleet Based Schedule Optimisation for Product Tanker 413

problem. There is no way of formulating the ship stability problem in terms

of linear inequalities.

Because a ship always has the ability of using ballast water to become sta-

ble enough, this has to be taken into account. Therefore the model contains

ballast water tanks and every possible combination of full and empty bal-

last water tanks is tested until stable conditions are reached. If this was suc-

cessful the program returns true to mark that the ship is compliant. If all

combinations are checked but the rules are not fulfilled for any of the bal-

last water tank combinations, the program returns false to mark that the

ship is not be compliant. All rules described in the following subsection are

well known and only a brief introduction shall be given here.

3.3 General aspects of ship stability In ship building the following approach to perform the stability evaluation

is required and commonly used. The 2008 IS-Code asks for certain proper-

ties of the lever arm curve (GZ curve) that have to be fulfilled at any situa-

tion at sea. The GZ curve is defined as the distance between the alignment

of the vector of the gravity force G and the vector of the buoyancy force B.

The GZ curve of one of the example ships used for the numerical experi-

ments described in section 4 is shown in Figure 2. While the ship is upright

we have GZ = 0. As soon as the ship heels, B moves towards the side of the

ship. Close to the favourable stable equilibrium the forces cause a torque

that pushes the ship back into the upright position. For a more detailed de-

scription see e.g. (Biran, 2003 and Tupper, 2013).

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414 Anisa Rizvanolli et al.

Figure 2: GZ curve of one of the example ships

In general, the gravity force G and buoyancy force B have to be at an equi-

librium when the ship does not move, viz. B = G, which is obvious due to

the principle of linear momentum. The principle of angular momentum

leads to the fact that the righting and the heeling lever arm have to be equal

at an equilibrium. Therefore, it is common practice to only look at the lever

arms for evaluating the ships hydrostatics. GZ depends on the heeling an-

gle and on the displacement of the ship. Because the lever arm is always

calculated for one particular displacement this dependency is neglected in

the following. The righting lever arm GZ (the dashed line in Figure 2) for

the light ship is defined by

GZ 𝜑 KN 𝜑 TCG cos 𝜑 VCG sin 𝜑 (25)

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Fleet Based Schedule Optimisation for Product Tanker 415

where TCG  is the transverse and VCG  the vertical distance of the center

of gravity of the light ship to the keel and

KN 𝜑 : TCB 𝜑 cos 𝜑 VCB 𝜑 sin 𝜑 (26)

with the transverse distance TCB and the vertical distance VCB of the cen-

ter of buoyancy in global coordinates. KN is also known as the cross curves

of stability when evaluated for different displacements. In Figure 3 this is

illustrated. There the center of gravity, marked as CG0  and CGϕ, as well as

the center of buoyancy, marked as GB0 and CBϕ, are shown. The index

stands for either no heeling angle (𝜑 = 0◦) or at some heeling angle (𝜑 > 

0◦).

Figure 3: Midship section

The position of the center of gravity of the ship has to be calculated accord-

ing to the loading status of the ship by:

TCG 𝜑 TCG 𝑦 , 𝜑 𝑉 𝜌 (27)

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416 Anisa Rizvanolli et al.

VCG VCG 𝑧 , 𝜑 𝑉 𝜌 (28) 

where 𝑦G,l (ϕ), 𝑧G,l (ϕ) are the transverse and vertical position of the center

of gravity of each single cargo hold, bunker and stores compartment and

ballast water tank. In the case of a free surface the position of the center of

gravity of the fluid within the tank changes due to the heeling angle of the

ship.

The effective righting lever arm GZ , which is the GZ of the light ship

corrected by the tank lever, as shown in Figure 2 (dashed/dotted line), is

calculated by

GZ 𝜑 KN 𝜑 TCG 𝜑 cos 𝜑 VCG 𝜑 sin 𝜑 (29)

which leads to the continuous curve in Figure 2. Thus the difference be-

tween the lever arm of the light ship (dashed line in Figure 2) at the draught

matching the displacement the ship in loaded condition and the effective

lever arm is a reduction caused by the cargo and in particular because of

the free surface. This effect is described in detail in e.g (Biran, 2003 and Tup-

per, 2013).

3.4 Criteria for intact stability The criteria which the 2008 IS-Code demand can be written as

GZ 𝜑 d𝜑∘

∘0.055mrad

GZ 𝜑 d𝜑∘

∘0.09mrad 

(30)

(31)

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Fleet Based Schedule Optimisation for Product Tanker 417

GZ 𝜑 d𝜑∘

∘0.033mrad 

GZ 𝜑∗ : max GZ 𝜑 with 𝜑∗ 25∘ 

GZ 𝜑 20∘ 0.3m

GM 0.15m 

where GM is the metacentric height of the upright floating ship. The crite-

ria 30 to 35 aim at ensuring a certain amount of energy a ship is able to take

up without risking to capsize. In addition to the mentioned criteria the

weather criterion has to be fulfilled. The basic idea is, that at the worst sit-

uation at sea a seafarer may think of, in terms of intact stability of the ship,

the ship shall withstand without being at risk of capsizing. For detail see

(Meier-Peter, 2012).

3.5 Application In the program the above described criteria are implemented and evalu-

ated for the MOERI Tanker KVLCC2 which is widely known in research. The

cross curves of stability as well as the displacements and other parameters

are provided in tabular form for heeling angles from 0◦ to 50◦ and displace-

ments from 141,000 t to approximately 321, 000 t. The values at an inter-

mediate state are interpolated linearly to avoid overestimation of the ca-

pability of the ship. For reasons of simplification the free surface of fuel oil

tanks and fresh water tanks is not taken into account. These tanks are small

compared to the ships size and do not have an impact on the ships stability

that is worth mentioning. But the free surfaces of a cargo hold that is not

(32)

(33)

(34)

(35)

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418 Anisa Rizvanolli et al.

fully filled or empty shall be taken into account. Therefore a simplified tank

model is chosen to determine the center of gravity when the ship heels (see

Figure 3). A quadrilateral shape of the transverse section is assumed. This

allows an analytical solution for the tank lever arms depending on the heel-

ing angle and the filling height of the tank. The integration of the resulting

GZ is realised by a simple quadrature and the interpolation. The criteria

may also be fulfilled not only at the departure condition but also at the ar-

rival condition to ensure that at every intermediate state reached during

the journey no unsafe condition occurred.

1. departure conditions with 100 % bunkers and stores

2. arrival conditions with 10 % bunkers and stores

The loading condition of each cargo hold of a ship within the fleet is set as

a result of the MILP. Then the criteria described above are checked for each

ship and a boolean true is returned if and only if all criteria at departure and

arrival are fulfilled.

4 Experiments and Analysis

As real world data was not available to us, we produced some artificial in-

stances. Due to the complexity of the problem and its exponential growth

we could only test the model on small instances. This means there is a small

number of orders and ports as well as a quite short time interval for few

ships that made the input parameters.

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Fleet Based Schedule Optimisation for Product Tanker 419

Table 1: Instance data

# ships ports orders time steps

1 2 3 6 15

2 3 4 110 30

3 3 3 7 15

4 3 3 7 15

5 3 3 7 15

The MILP program is implemented in C++ and the GUROBI (8.1) solver (aca-

demical license) is used for solving the problem. We ran the solver on a

quad-core processor (i5-6600 CPU @ 3.30GHz) with 16 GB random access

memory. The stability calculations are also implemented in C++ and com-

municate directly with the MILP. Whenever a MILP solution is found, it is

passed to a function that returns true or false depending on whether the

stability constraints are satisfied.

All ships in all five problems are of the same type. They have 15 tanks with

a volume of approximately 10, 000 litres or 15, 000 litres. They are arranged

in a 3 X 5 grid, which is important for both the ship stability constraints as

well as the adjacency constraints. The tank volumes were discretized into

units of volume 5, 000 litres to reduce computation time.

Instance 1 was constructed to have a feasible solution with respect to the

ship stability constraints. It was solved in 5 seconds. The optimal solution

completed all orders.

Instance 2 turned out to be much harder to solve than the first one. After 20

minutes the solver found a solution, which completes 7 out of 10 orders.

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420 Anisa Rizvanolli et al.

After two hours in total, we stopped the program. It had not found a better

solution in the meantime. The MIP gap was at 44%.

Instance 3 to 5 were the same with minor differences. Compared to instance

3, in instance 4 we set γz,y = 0 and δz,y = 1 to remove all good type re-

strictions. Instance 5 in turn was the same as instance 3 except that we mul-

tiplied all densities of good types by 0.7, which relaxes the ship stability con-

straints. The lighter the goods, the smaller is the effect on the stability. Dur-

ing the two hours in which we let the solver tackle this problem, its best

solution was found already after about half an hour. In the remaining time

the solver was able to reduce the MIP gap to 17.7%. Removing the good type

restrictions did not help the solver find a solution. In fact, because it in-

creased the number of solutions, it only found its best solution after 36

minutes. The gap after two hours was again 17.7%. Finally, for instance 5

the changes we made significantly affected the solver. The solution that

was also found for instances 3 and 4 was already found after 5 minutes. Half

an hour later it found a slightly better solution with respect to the number

of journeys and loading operations. After a total of two hours it had de-

creased the MIP gap to 17.5%.

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Table 2: Results with stability

# obj. value * of best solution time (s)

1 589 5

2 681 1113

3 587 1899

4 587 2199

5 588 1992

Table 3: Results without stability

# obj. value * of best solution time (s) Intermediate result

1 589 5

2 981 92 683 after 38 s

3 684 213 588 after 19 s

4 686 108 589 after 7 s

5 684 213 588 after 19 s

As a comparison we also ran the solver on these instances without stability

constraints, which show significant differences in run time and in the best

solution found within two hours. The objective values can be interpreted as

follows: Each completed order adds 100 to the value and each loading op-

eration and ship cruise subtracts 1 from it, i.e. we set Or = 100, μsp = 1 and λspq = 1.

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As shown in Table 3 and 4, adding the stability constraints significantly in-

fluences the run time of the program. Instance 1 was solved in both cases

very fast. But as soon as the tested instance got more complicated, the

stability constraints led to a big difference in the run time. This can be ex-

plained by the number of solutions of the MIP which have to be tested in

order to find a feasible solution for the stability constraints. In the case that

the progam finds a feasible solution with objective value k, it has to test

whether this solution satisfies the stability constraints. If not, the pro-

gram tries to find a different assignment to achieve an objective value k. If

the stability constraints are excluded, the program directly tries to improve

the objective value, which means that significantly fewer feasible solutions

for the MIP have to be found. This also explains why the best solution with

stability constraints takes way longer to find than to find a solution with this

objective value without stability constraints. Unfortunately real world

problems are much larger than our test instances. To conduct more mean-

ingful numerical experiments, we need to further develop our approaches.

5 Conclusions and Future Work

This work shows that mathematical models and methods can improve the

planning process significantly and give well-founded decision support for

planners. With this first model it has been shown that the integration of non

linear requirements as the ship stability check to common MILP formula-

tions of the cargo scheduling problems is possible. This inclusion makes the

model even more realistic and enables calculations of scenarios that are

very time consuming done in a manual way. The constraints of this MILP

formulation link many different aspects of the cargo scheduling problem

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Fleet Based Schedule Optimisation for Product Tanker 423

making it on the one side quite complex to solve real-world instances ex-

actly and more realistic on the other side. This formulation is a first draft

and further analysis of the numerical results need to be made in order to

identify performance potentials. The examination of appropriate algo-

rithms and methods, that take advantage of special problem structure and

properties, is missing and is considered to be one of the main factors with

a big impact in the calculation time. The development of appropriate heu-

ristics is also one key aspect to be taken into consideration for future work.

Beside the performance issue the model itself can be made even more re-

alistic by considering the speed as a decision variable and fuel consump-

tion in dependence to loading. From a ship design point of view, in particu-

lar for bulk carriers of liquid or solid cargo, the longitudinal strength is a

problem worth mentioning. In this work the view on mechanical require-

ments is neglected but for real world application an evaluation also has to

be applied to the algorithm. Modeling the complex problem of cargo sched-

uling as realistically as possible and adapting methods for solving real data

instances is one of the main future challenges.

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